Fine-tuning excludes the Multiverse Hypothesis and confirms Coherent Cosmology, Eddington's and String Theories F. M. Sanchez, M. Grosmann, D. Veigel, R. Veysseyre, C. Bizouard, D. Gayral, L. Gueroult

Fine tuning relations using Eddington number 137 and the central mathematical constants e, π, γ connect with the Fermi coupling constant, the muon and tau masses, and the G value. The latter is compatible, within 10-7 with the G value deduced from the Coherent Cosmology, and within 10-9 to the magnetic moment. The formula simplify by introducing a super Euler - constant, and show a 9D reduction to 3D, confirming Superstring Theory, while c-free analysis confirms the Topological Axis and the tachyonic Bosonic String Theory, with emphasis on a 30 dimension Space, connected through triple Bott reduction with a massive Gluon in 6 dimension space. The 26 sporadic groups are tied to the three microphysics coupling constant and the gauge group dimension 496, whose square give the Higgs boson/Electron mass ratio. The Baby-Monster is directly tied to the observable . The Monster Group, whose cardinal order is directly tied to Rydbergh energy (10-6) appears in the c-free analysis giving the overall periodicity of a Deterministic Grandcosmos Computer. This confirms the exclusion of the Multiverse hypothesis and Eddington's rehabilitation.

The Eddington's number 137 has a fundamental property: it appears as a Monster Prime in the series of the maximal primes appearing in the numerator of the harmonic series: 3,11,5,137,7,11, showing a symmetry between the 11 supergravity dimensions and the 4 of space-time:

137 = 11² + 4²

Since Riemann series are tied to the prime number distribution, it is strange that mathematicians have not point out the primes appearing in the Harmonic series, since it is the single pole. It seems that the basic precept 'all occurs in the pole' was forgotten in this case. As ancient Egyptian used only fractions of type 1/n, they were certainly aware of this particular number 137. Indeed the fifth harmonic series s5 = 137/60 appears in the Ptolemaic approximation for π: 377/120 = 2 + s5/2. Recall that the electrical constant characterizes the force ħc/al² between two l - distant elementary charges, appearing central in Atomic and in many fine-tuning relations [2]. It is strange that physicists focused on only one property, the appearance of its fifth power in the hyper-fine spectra, and call its inverse the 'fine-structure constant'. It is strange also that Eddington's Theory was rejected as soon as a appeared to be different from 137. The present article shows that 137 plays a central role in fine-tuning analysis. Indeed a simple computer study shows that 137 and the Euler-Mascheroni constant γ enter the relation:

2 3 GF/mec ≈ (πƛe/2γ137a)

3 -62 where ƛe ≡ ħ/mec is the reduced electron radius, and GF ≡ (ħc) /EF ² ≈ 1.4358509(7) ×10 Joule × 3 2 m is the Fermi coupling constant, corresponding to the Fermi energy EF = Fmec ≈ 292.806161(6) 2 GeV ≈ 573007.33(25) mec [3]. 2 -7 Admitting the above relation, this defines F = EF/mec ≈ 573007.3652, inside its 2.5 10 indetermination. Now the latter enters another fine tuning relation, induced by the Kotov Coherent 2 Cosmic Oscillation [4], implying the muon, and Hydrogen masses: EF/mec ≈ 2 3 -8 mµ √(mpmH)/ame . This corresponds to a muon mass 206.7682869 me, inside its 2 × 10 measurement range. The a-priori probability of such correlations are estimated, by looking for the number of solutions obtained by extending the indetermination range by a factor 106, keeping a maximal exponent to 3, to be respectively 2 × 10-5 and 10-6. From the Koide relation [5], one gets the corresponding mass ratio tau/electron: τ ≈ 3477.441701, see below. Fine-tuning research also show that the most elegant formula for the strong coupling constant is [4]

3/2 f ≡ aw/2π(pH) ≈ 8.434502892 while the CODATA value is badly defined fCODATA ≈ 1/0.1181(11). With 496, the dimension of the gauge group of type 1 string theory, one notes, to 0.05 % :

F/af ≈ 496

The importance of this Central Relation between the three microscopic coupling constant will be confirmed later, in connexion with the sporadic groups. Note that the mass ratio scalar Higgs Boson/electron is close to:

s ≈ 196² corresponding to 125.7 GeV, while the experimental value is 125.09(24) GeV. Now, the three first terms of the Combinatorial Hierarchy [6] are the connected Mersenne numbers 3, 7, 127, whose sum is 137, so giving the first order of the electrical constant. The following (final) term 2127-1, the most famous prime in number theory, gives the first order of the gravitational coupling, whose optimal form results from the gravitational Hydrogen molecule model [7] aG = ħc/GmpmH. Computer analysis shows the following symmetrical expression for the deviation, with n ≈ 1838.68366089(17) the neutron/electron mass ratio:

127 1/2 4 3 (2 /aG) ≈ F(a/π) (γ /4n) this corresponds to

G ≈ 6.675453818 × 10-11 kg-1m3s-2 compatible within 10-7 with the value deduced from Coherent cosmology [7]. It is compatible with the well-elaborate 10-5 BIPM measurement [8], but at several sigmas from the Codata value, since the later is the mean between discordant measurements. A secular variation of G with a terrestrial period has been suggested, but not consistent with the phases. The formula exhibits a symmetry between a and π. In this respect one notes that, to 2 ×10-7: a² ≈ 137² + π², and that, by analogy with the Euler formula: i-ilni = exp(π²/2) ≈ √(a+2), and the first -ilni -9 development term of de, with u = i √a -1, one obtains a within 3 × 10 :

a ≈ u - 1/2πu = 137.035999548

Moreover, one obtains a value a ≈ 137.035999119 compatible with measurement a ≈ 137.035999139(31) in :

2 ln137/ln(a/137) ≈ (2+135/de) with de ≈ 1.00115965218091(26), the anomalous electron magnetic moment, meaning the ratio a/137 shows a type of musical role. Now, with the above value of G, the mass ratio proton-electron p ≈ 1836.15267389(17) and the mass ratio Hydrogen-electron H ≈ 1837.15264726(17), the computer indicates a 10-9 correlation: 127 1/2 3 (2 /aG) ≈ de(H/p)

This writes: 1/3 (aF/πde ) ≈ 4πn'/γ a where n' = nH/p is the principal value of the neutron mass by respect to the electron effective mass in the Hydrogen atom. Note the 0.12 % proximity of the above number with the fifth fractional development of π: 292.6345909, itself approaching n/2π to 3.4 10-6. By introducing a super-Euler constant : Γ = γ a/π the relation giving GF becomes much simpler:

2 3 GF/mec ≈ (ƛe/137×2Γ ) and the above relation giving aG shows a double form:

127 1/2 2 3 -3 3 -1 (2 /aG) ≈ (a/π)(nF/137 Γ ) ≈ (a/π)((4n/Γ) /F) corresponding to a dimensional reduction, by eliminating 137, from 9D and 6D to 3D, which could be associated to Superstring theory, where the equations are coherent only if space has 9 dimensions, and if the 6 supplementary dimensions are fold on very small distances. The above tau number enters the following much symmetrical relation, to 1.5 ×10-6:

(τ²Γ / a²)3 ≈ 4pnH (496(H-p)/137)4 calling for further analysis. Such relations are unbelievable for reductionist people, arguing that since neutron is composite, it cannot enter simple relations. The same argument is presented for the theoretical dependence of a with other constants g and g'. But Coherent Cosmology [7] shows that the holistic point of view is more pertinent than the classical reductionnist one. Moreover, both the number 137 and the term aa appear in musical number analysis [9], confirming that a is an optimal calculation basis. Such transcendental formulas are surprising for theorists which forget that beauty is the root of Science. In particular, the famous Lucas-Lehmer primality test uses the series of whole numbers Nn+1 = Nn² – 2, starting from N = 4 = u3 + 1/u3, with u3 = √3 + 2, belonging to the Diophantine generators un = √n + √(n+1). One shows that Nn ≈ u3^(2^q), and for q = 9:

64 a u3^(2^9) ≈ (2×(a² + 2√µ)) ≈ a

defining a to 39 ppm, where µ is the mass ratio muon/electron and the main term 2a² = mec²/ERyd is tied to the Rydbergh energy's principal value ERyd whose ratio with the Planck energy is closely related to the Monster group cardinal order, to 1.5 × 10-6:

-1///2a 5 2 OM e ≈ (EP/ERyd)² = ħGc /ERyd

See below the incredible other properties of OM. Also, with the Pell-Fermat generator u1 = 1 + √2:

a 8 a ≈ u1^(3×(2 - 1))

a defining a to 0.3 ppm. So the number a establishes a connexion between u1 and u3, two of the simplest arithmetics generators. This opens a new research in pure mathematics. The number π appears as a calculation basis in the Riemann series of even order. So, it is natural to consider the ratio π/e. Note that, with 34, the number of primes leading to 137 (by counting 1 as a prime number) the Bohr atomic radius writes, to 1.4 × 10-6:

34 rH = a ƛe(1+1/p) ≈ (H-p) (π/e)

This leads to the discovery of the incredible relation:

3 1/2 p (2/3)(2aG) ≈ (a/137) (π/e)

-8 which confirms the above G value to 4 × 10 . Now, since aG is tied to the Universe horizon radius by 2aG = R/ƛe, the factor 2 coming from the number 2 of Hydrogen atoms in the Hydrogen molecule 2 [6], the Universe volume enters this formula, which favors R = 2aGƛe = 2ħ /GmpmHme ≈ 13.812 2 3 3 Glight-year, by respect to R' = 2ħ /GmN = RpH/a where mN = ame is the Nambu mass, central in Particle Physics, and which look simpler since R'/2 is obtained by elimination of c between the electron classical radius and the Planck length. It is shown that this corresponds to a global coherence condition in the Critical Universe considered as a sequential C-connected Universe, where C is a tachyonic speed, far exceeding c. It has been assumed that R' is the holographic trace of a Grandcosmos behind, whose radius exceeds R by the ratio C/c, and whose volume, with length a unity the bare Hydrogen radius rH = a ƛe exhibits the central term a [6]:

3 a p VGC/rH ≈ a /π ≈ (1/ln2)

This sustains the hypothesis that the Grandcosmos is the source of the Cosmic Microwave Background, as confirmed below. This is the Absolute Frame that Relativity Theory is unable to define, so cannot explain what is a Galilean frame, as it cannot really explain Foucault pendulous or Sagnac experiment. Now, the ratio 2R/R' shows the property, to 4 ppm:

-3²×5×7 2×3×5×7 R/ƛe ≈ e (2R/R') implying a Space of 2 × 3 × 5 × 7 = 210 dimensions, predicted long ago by the Holic Principle [7]. Now, the ratio lnp/lna is close to this canonical ratio 2R/R' [3], so

a2^a3 ~ p^p2 ~ P2s showing a combinatorial geometric significance, where P is the Weyl's ratio mP//me. The essential critical condition R = 2GM/c², where M is the total equivalent Universe mass is explained by a 1D-2D-3D Holographic Resonant Conservation [7] implying the Universe wavelength d = ħ/Mc, smaller than the Planck length by a factor about C/c ≈ 1060. This means the Machian relation:

4/ M = mP mpmHme ≈ (10/3)NEd mn ~ Ev/C²

/1/2 256 where mP = (ħc/G) is the Planck mass, NEd = 136 × 2 the Eddington Large Number, and Ev the vacuum quantum energy, so resolving the central enigma in present-day physics: why the vacuum quantum energy is about the 10120 times the Universe one. From the critical relation R = 2GM/c² and the classical energy of a homogeneous ball E = 3GM²/5R, one obtains E = (3/10)Mc². So this trivial gravitational factor 10/3 corresponds to the misleading 'black energy' of the standard cosmology which needs complete re-interpretation. The fact that Eddington predicted correctly the effective mass 3M/10 (but in term of neutrons instead of Hydrogen) is probably the most remarkable prediction of all times. Moreover, the antimatter problem is resolved by admitting that Universe is subjected to a 10104 Hz matter-antimatter oscillation. Black Matter would be simply a quadrature oscillation [7]. In their famous paper [2] Carr and Rees recognize that the three constants a, p and aG suffice to explain the basic features of , , planets and the everyday world. But they absurdly call -40 Gmp²/ħc the 'gravitational fine structure constant', as if a correction of 10 would have any sense. They add: 'However, one of the scale in Fig.1, that associated with the Universe, cannot be explained directly from known physics: it is apparently a coincidence that the present age of the -1 Universe is of the order of αG times the electron timescale. This is misleading for the following reasons: firstly, in their Fig. 1, only lengths are considered, not times. Secondly, what is measured directly in the recession is a length, that of the Universe horizon radius. Thirdly, Eddington's theory gives a statistical explanation (using this third pillar of physics, forgotten in favor of Relativity and Quantum Physics). Finally, the length-formula is simpler than the time-formula, since c disappear (this was not noticed by Eddington, since he worked with c = 1). 8 However the Carr and Rees paper has the merit to underline the relation: aG ≈ W , where W is the mass ratio W boson/electron. Now, in its 10-3 experimental indetermination, one observes the following more symmetrical relation, implying also Z, the neutral weak boson, confirming the Topoligical Axis [7]:

4 R/ƛH ≈ (WZ)

Note that R/2ƛH plays a central role in the definitive resolution of the so-called Large Number Problem, which shows that the Eddington 'reference mass' is the effective electron mass [7]. This milits for a Single-Electron Cosmology, or the Black Atom Model, which precises the rough estimation a ~ lnaG, considered correctly as central by Carr and Rees. Now, considering that the celerity c is inadequate in Cosmology is exactly one must expect in a Coherent Universe, because c is far too small to interconnect a so vast Universe. One of the author deduced the formula R ~ ħ2/Gm3, where m is an atomic mass, in its 3 first minutes of its sabbatical year 1997-98, by applying the elementary 3-fold dimensional analysis, simply by discarding c as being a non pertinent cosmic speed. This has not been done before because theoreticians foolishly put c = 1 in the equations, mixing Length and Time in the all-Relativity spirit, forgetting the warning of Poincaré, the true discoverer of Relativity, against a too close identification Length-Time. Poincaré also claimed that Cosmology cannot be based on differential relations, because as the Universe is unique, one cannot define integration constants. More precisely, it is shown [7] that Relativity is a local concept, not applying at the Cosmic Scale: indeed the non-relativistic kinetic energy of the galaxies, receding with the simplest exponential law of time constant R/c, is precisely the above 3Mc²/10. Note that this exponential law is equivalent with an invariant 'cosmological term' added in the GR equations. This means that, at a large scale, matter is repulsing other matter with a force proportional to length. This is not necessarily a reversal of gravitational law at large distance, it suffices that 'inverted repulsive matter' is, at cosmic scale, uniformly distributed. This would explain the large bulle-structure of the galaxy groups distribution. In a Permanent Cosmology, such a galaxy recession is necessary, to obeys the Second Principle; otherwise all the stars would have vanished long ago. 3 2 3 In resume, with m ≈ mpmHme the c-free analysis L{ħ,G,m} = ħ /Gm gives precisely half the Universe Horizon radius. Now the associated time is:

T{ħ,G,m} = ħ3/G2m5 ≈ 5.530 × 1057 s by respect to the electron-time te = ħ/mec², one observes:

30/4 T{ħ,G,m}/te ≈ f(30) = exp(2 ) this is precisely (4 %) the lacking essential point n = 30 in the Topological Axis, for which the special string bosonic value n = 26 corresponds to the Universe, apart a factor 6:

3 5 R/ƛe ≈ (2π²a ) ≈ f(26)/6 precise to 0.056 % and -0.065 %, where 2π²a3 is the area of the 4-sphere of radius a. Now f(30) = f²(26), so implies a30, meaning a 30D Space is really involved: the rehabilitation of Bosonic String Theory [6] by the Topological Axis is confirmed. It has been discarded because it induce tachyons. Of course, in Coherent Cosmology, this is an essential advantage. Note that n = 30 is the single solution of a Perimeter Equal to Area non-decomposable pythagorician triangle (12, 5, 13). The only other one, but decomposable, is the triangle 6,8,10, with perimeter = area = 24, which is the number of transverse dimensions in String Theory. The Topological Axis clearly shows the Bott- Cartan periodicity ∆n = 8: 30-8 = 22, corresponding to the X (strong) boson, 22-8 = 14, corresponding to the intermediate (weak) boson, 14-8 = 6, corresponding to a massive gluon. Now, comparing T with the Kotov Non-Doppler Cosmic Oscillation period tcc ≈ 9600.60(2) s, one gets, to 0.8 %:

T{ħ,G,m}/tcc ≈ OM/√2 where OM is the cardinal order of the Monster Group, the larger of the 26 sporadic groups, The product of the 26 sporadic group orders connects (0.16 %) with the Grandcosmos radius RGC:

3 (R/R') Πtot ≈ (RGC/lP)

This would mean that the totality of the 26 sporadic groups are implied in the determination of the physical parameters, called 'free parameters' in the Particle Physics Standard Theory, whose number is precisely about 26. Note that, generally, the number of dimensions may be interpretable by a number of parameters. The product of the orders of the 20 groups of the 'happy family' is, to 0.015 %:

a (R/R') Πhappy ≈ a

The product of the six pariah groups enters, to 0.02 % and 0.2 %:

Πpariah/OB ≈ (2τ/p)R/ƛe

1/20 Πparia ≈ F/a where τ/p is the tau/proton mass ratio, whose importance was predicted by Eddington, and F/a = v/c corresponds to the redshift periodicity v = 72 km/s, corresponding to about 1 million ligth-years, typical of a galaxy group. Eliminating c = ħ/meƛe this gives

ħ ≈ mF re v showing v to be a kind of speed quantum in a clear quantum resonance condition involving the Fermi mass and the electron classical radius re = ƛe/a, so exhibiting a quantum property of the vacuum. Moreover : 1/20 (Πheur/OM) ≈ f

So the above Central Relation F/af ≈ 496 receives an illustration trough the families of sporadic groups. The Monster itself shows the following intrinsic property, to 8 × 10-6: lnOM/lnlnlnlnOM ≈ 2 × 137

The product of orders of the Monster and the Baby is tied to the number of photons in the is, to 0.1 % :

nph ≈ OMOB √(R'/R)

The number of photons in Grandcosmos is presented below. This clearly favors Permanence by respect to Evolution. This is directly confirmed by the following evidence of the invariance of the background temperature θCMB ≈ 2.7258 Kelvin. Looking 3 for the mass of a black hole having an Hawking temperature θH = ħc /8πkGmH equal to θCMB , one finds, to 0.7 %:

mHkg ≈ me (R/R') OM /4π

which writes, by suppressing the 4π factor, with ƛCMB = ħc/kθCMB :

2 (R/R') OM ≈ ƛeƛCMB/2lP

This confirmation of Temperature Invariance comes in addition to those already noted [7], in 2 2 particular the fact that the 1D-2D Holographic writing of ħ /GmemB traduces an electron-baryon symmetry, while the extension to 3D exhibits the CMB wavelength ƛCMB. Moreover, this means that c-free analysis starting from ħ,G and the energy kθCMB is close to ƛH. Moreover, the reduced Wien constant ϖ = 5(1−e−ϖ) ≈ 4.965114245 defining the Wien wavelength λWien = λ/ϖ = hc/ϖkT enters a 40 ppb relation with n:

n ≈ (ϖ(π/2)2)3

Now, the radius R' shows, to 0.1% :

a 4π(R'/λWien)² ≈ e

The perfect holographic form of this relation casts a serious doubt on the true incoherence of a thermal radiation, as in the problem of the information conservation in a black hole. One observes also that the half-volume of proton shows:

3 a (2π/3)(rp/lP) ≈ e

3 meaning an intervention of the cube lP , while the standard holographic principle uses only the area ϖ lP². Moreover, one observes that a ≈ e – 2π, suggesting a to be a trigonometric line. Indeed cosa ≈ 1/e. So, to 65 × 10-9:

a ≈ 44π – Arccos(1/e)

Another important Planck Law's number is the Riemann series ξ(3) ≈ 1.20205691, or 'Apéry constant', with no analytic expression, but which gives the photon density16πξ(3)/λ3, where λ = hc/kBθ. The computer indicates, to 1.6 ppm:

√a ≈ (16ξ(3))3/ϖ4

With our precise value [6] θCMB ≈ 2.725828 Kelvin, the number of photons in the visible Universe is 3 87 nph = (4π/3)(kBθΧΜΒR/hc) ≈ 3.8400458 × 10 , while the equivalent neutron number is nn = (10/3) × 256 79 3 6 60 136 × 2 ≈ 5.2492414 × 10 . With the ratio RGC/R = C/c = P pH/a ≈ 6.9454957 × 10 , the 3 number of photons and equivalent neutrons in the Grandcosmos are respectively Nph = nph(C/c) ≈ 3 exp(621.949984) and Nn = nn(C/c) ≈ exp(603.841903). One observe that the mean obeys:

5 n/3 √(Nph Nn) ≈ (n/6π ) e precise to 6 ppm on a number with 267 decimal digits. Now, another decisive point is the invariance of the Universe mean mass density, which is tied -27 -3 to G by the critical condition: ρc = 3c²/8πGR² ≈ 9.41198 × 10 kg m . Inserting instead the Fermi Constant in the c-free analysis leads to:

4 3/2 5/2 57 T{ħ,ρc,GF} = ħ /ρc GF ≈ 5.4829 10 s i.e. about the same time as above, which connects even more closely (0.04%) with OM/√2. This is interpretable as the lacking element in Coherent Computing Cosmology: the overall periodicity of History, in a completely deterministic Diophantine Grandcosmos. These are striking examples of the extreme precision of the fine tuning between physical parameters, to be compared with the large imprecision of Anthropic Principle arguments. The present-day physics community is divided in two camps, a minority who believe in a Single Final Theory, and a large majority having abandoned hope and believing seriously in the Multiverse conundrum. The present article settles the debate. Another type of separation exists: a minority think Physics and Mathematics are unified, while a majority separate the two domains (so separating also Biology). The present article shows that the former are right: physical constants are mathematical constants, so the present-day mathematics are still in enfancy, not realizing that the discovery of sporadic groups is a crucial discovery. In particular, we have clearly shown that Grandcosmos is a computer which uses optimal physico-mathematical constants as calculation basis and that they are present in DNA [7]. The present article show definitely the liaisons with π, e and γ, and rehabilitate String theories, also foolishly abandoned by a majority. There is also the Determinism separation, a majority believing seriously that 'God plays dices', which is contradictory with the above Computing Principle. The c-free analysis gives simply and directly the Large time periodicity of an all-deterministic Grandcosmos, as it gives in an elementary calculation the visible Universe horizon radius, in a formula which was present for a century in text-books: the limit of a radius when the number of atoms reduce to 1 [7]. This is tied to the application of the exclusion principle that Eddington dared to apply in cosmology. For this reason he was declared 'crakpot' and his theory discarded by a majority. Fortunaly it is now recognized the large theoretical advance of Eddington [10][11], but without mention that he had predicted the tau fermion, 30 years before its surprising discovery, calling it Heavy Mesotron,with a right order of mass [1] .

References

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[3] P.J. Mohr, D.B. Newell, and B.N. Taylor in arXiv:1507.07956 (2015)

[4] C. Bizouard, Les Oscillations Cosmiques. Conférence au Collège de France (2004) [5] Koide Y., Fermion-Boson Two-Body Model of Quarks and Leptons and Cabibbo Mixing Lett. Nuovo Cimento 34, 201 (1982)

[6] Bastin T. and Kilmister C.W., Combinatorial Physics (World Scientific, 1995).

[7] F.M. Sanchez. Coherent Cosmology Vixra.org,1601.0011. Springer International Publishing AG 2017. A. Tadjer et al. (eds.), Quantum Systems in Physics, Chemistry, and Biology, Progress in Theoretical Chemistry and Physics 30, pp. 375-407. DOI 10.1007/978-3-319-50255-7_23.

[8] Quinn T, Speake C, Parks H, Davis R. 2014 The BIPM measurements of the Newtonian constant of gravitation, G. Phil.Trans. R. Soc. A372: 20140032.

[9] F.M. Sanchez. Musics and Science. viXra:1806.0341

[10] Durham I.T. 2006, Sir and the Foundations of Modern Physics arXiv:quant- ph/0603146v1 p.111.

[11] Salingaros N., Some Remarks on the Algebra of Eddington's E Numbers. Foundations of Physics, June 1985, Volume 15, 6, pp 683–691.